Let $\mathcal X$ be a regular scheme, flat and proper over the ring of integers of a $p$-adic field, with generic fiber $X$ and special fiber $\mathcal X_s$. We study the left kernel $Br(\mathcal X)$ of the Brauer-Manin pairing $Br(X)\times CH_0(X)\to \mathbb Q/\mathbb Z$. Our main result is that the kernel of the reduction map $Br(\mathcal X)\to Br(\mathcal X_s)$ is the direct sum of $(\mathbb Q/\mathbb Z[\frac{1}{p}])^s\oplus (\mathbb Q/\mathbb Z)^t$ and a finite $p$-group, where $s+t= ρ_{\mathcal X_s}-ρ_X-I+1$, for $ρ_{\mathcal X_s}$ and $ρ_X$ the Picard numbers of $\mathcal X_s$ and $X$, and $I$ the number of irreducible components of $\mathcal X_s$. Moreover, we show that $t>0$ implies $s>0$.
Varietes de descente, gerbes et obstruction de Brauer-Manin (Descent varieties, gerbes and Brauer-Manin obstruction)
我们对布劳尔-马宁的一系列障碍物进行了界定，就像在各种情况下，全球部分的存在缺陷一样。Ceci nous conduit à une généralisation de la dualité de Tate-Poitou au cas non-abélien.我们找到了一种方法来关联到一个gerbe定义在数域上的Brauer-Manin阻塞，它的措施，作为品种，缺陷的存在一个整体瓣。这导致我们在非阿贝尔的情况下推广的Tate-Poitou对偶。
We investigate the question of whether the existence of a family of local zero-cycles of degree $d$ orthogonal to the Brauer group implies the non-emptiness of the Brauer-Manin set for certain varieties. We provide various examples of Brauer-Manin obstruction to the existence of zero-cycles of appropriate degrees.
We generalize the results of Skorobogatov and Zarhin considering the commutativity of Brauer groups (and Brauer-Manin sets) with taking product of two varieties, by relaxing the condition that varieties are projective.
We prove that, on a sufficiently general diagonal quartic surface, there is a non-trivial Brauer group but no Brauer-Manin obstruction to the existence of rational points.
We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the infinite etale Brauer-Manin set. The key new ingredient is the arithmetic of singular projective curves.
We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle.
We establish Grothendieck's section conjecture for an open subset of the Reichardt-Lind curve, and introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension.
We construct a conic bundle over an elliptic curve over a real quadratic field that is a counterexample to the Hasse principle not explained by the étale Brauer-Manin obstruction. We also give simple examples of threefolds with the same property that are families of 2-dimensional quadrics, and discuss some other examples and general properties of the Brauer-Manin obstruction.
Let $k$ be a number field. We construct homogeneous spaces of $SL_{n,k}$ with finite nilpotent non-abelian stabilizers for which the Brauer-Manin obstruction does not explain the failure of strong approximation (resp. the failure of the integral Hasse principle).
For a smooth proper variety over a $p$-adic field, the Brauer group and abelian fundamental group are related to the higher Chow groups by the Brauer-Manin pairing and the class field theory. We generalize this relation to smooth (possibly non-proper) varieties, using the motivic homology and the tame version of Wiesend's ideal class group. Several examples are discussed.
The Brauer-Manin obstruction is used to explain the failure of the local-global principle for algebraic varieties. In 1999 Skorobogatov gave the first example of a variety that does not satisfy the local-global principle which is not explained by the Brauer-Manin obstruction. He did so by applying the Brauer-Manin obstruction to étale covers of the variety, and thus defining a finer obstruction. In 2008 Poonen gave the first example of failure of the local-global principle which cannot be explained for by Skorobogatov's étale-Brauer obstruction. However, Poonen's construction was not accompani…
We use Brauer-Manin obstructions to explain failures of the integral Hasse principle and strong approximation away from infinity for the equation x^2+y^2+z^k=m with fixed integers k>=3 and m. Under Schinzel's hypothesis (H), we prove that Brauer-Manin obstructions corresponding to specific Azumaya algebras explain all failures of strong approximation away from infinity at the variable z. Finally, we present an algorithm that, again under Schinzel's hypothesis (H), finds out whether the equation has any integral solutions.
Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\mathbb Q$, and let $\ov X$ and $\ov Y$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of $\Br(\ov X)\oplus \Br(\ov Y)$ has finite index in the Galois invariant subgroup of $\Br(\ov X\times\ov Y)$. This implies that the cokernel of the natural map $\Br(X)\oplus\Br(Y)\to\Br(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer-Manin set of the product…
We study the Brauer-Manin obstruction to the existence of zero-cycles of degree $d$ on certain classes of varieties over number fields. We generalise existing results in the literature and prove some results about fibrations over the projective line, where the geometric Brauer group of the generic fibre is not assumed to be finite. The idea is to assume that the Brauer-Manin obstruction to the Hasse principle is the only one for certain fibres and then deduce analogous results for zero-cycles.
In this paper, we study the properties of weak approximation with Brauer-Manin obstruction and the Hasse principle with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. We assume a conjecture of M. Stoll. For any nontrivial extension of number fields $L/K,$ we construct two kinds of smooth, projective, and geometrically connected surfaces defined over $K.$ For the surface of the first kind, it has a $K$-rational point, and satisfies weak approximation with Brauer-Manin obstruction off $\infty_K,$ while its base change by $L$ does not so off $\infty_L.$ F…
We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over $\bbQ$ such that $\Br(S)/\Br(\bbQ)$ is of order two or four. This covers the vast majority of the cases when this group is non-zero. Our approach is to associate a Brauer class with every Galois invariant double-six. We show that all order two Brauer classes may be obtained in this way. We also recover Sir Peter Swinnerton-Dyer's result that $\Br(S)/\Br(\bbQ)$ may take only five values.
Recently Dasheng Wei proved that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on some fibrations over the projective line defined by bi-cyclic normic equations. In the present paper, we prove the exactness of the global-to-local sequence for Chow groups of 0-cycles of such varieties, which signifies that the Brauer-Manin obstruction is also the only obstruction to weak approximation for 0-cycles of arbitrary degree. Our main theorem also generalizes several existing results.
In this paper, we study the property of weak approximation with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. For any nontrivial extension of number fields L/K, assuming a conjecture of M. Stoll, we construct a smooth, projective, and geometrically connected surface over K such that it satisfies weak approximation with Brauer-Manin obstruction off all archimedean places, while its base change to L fails. Then we illustrate this construction with an explicit unconditional example.
In 2011, Vàrilly-Alvarado and the last author constructed an Enriques surface $X$ over $\mathbb{Q}$ with an étale-Brauer obstruction to the Hasse principle and no algebraic Brauer-Manin obstruction. In this paper, we show that the nontrivial Brauer class of $X_{\bar{\mathbb{Q}}}$ does not descend to $\mathbb{Q}$. Together with the results of Vàrilly-Alvarado and the last author, this proves that the Brauer-Manin obstruction is insufficient to explain all failures of the Hasse principle on Enriques surfaces. The methods of this paper build on the ideas in several recent papers by the last autho…
Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3\nmid[L:k]$. Therefore, the conjecture of Colliot-Thélène and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement…
This note gives an explicit example of transcendental Brauer-Manin obstruction to weak approximation. It has two features which the only previously known example of such obstruction did not have: the class in the Brauer group which is responsible for the obstruction is divisible, and the underlying algebraic variety is an elliptic surface (a K3 surface).
We analyze the Brauer-Manin obstruction to rational points on the K3 surfaces over $\mathbb{Q}$ given by double covers of $\mathbb{P}^2$ ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer-Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou …
We show that, for every integer $1 \leq d \leq 4$ and every finite set $S$ of places, there exists a degree $d$ del Pezzo surface $X$ over ${\mathbb Q}$ such that ${\rm Br}(X)/{\rm Br}({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z}$ and the Brauer-Manin obstruction works exactly at the places in $S$. For $d = 4$, we prove that in all cases, with the exception of $S = \{\infty\}$, this surface may be chosen diagonalizably over ${\mathbb Q}$.
Let $\mathbb{F}$ be a finite field and $C,D$ smooth, geometrically irreducible proper curves over $\mathbb{F}$ and set $K = \mathbb{F}(D)$. We consider Brauer-Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve $C \otimes_\mathbb{F} K$. In particular, we show that Brauer-Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of $D$ is less than that of $C$. We also show that we can identify the points corresponding to non-constant maps $D \to C$ using Frobenius descents.
We relate the Brauer group of a smooth variety over a p-adic field to the geometry of the special fibre of a regular model, using the purity theorem in étale cohomology. As an illustration, we describe how the Brauer group of a smooth del Pezzo surface is determined by the singularity type of its reduction. We then relate the evaluation of an element of the Brauer group to the existence of points on certain torsors over the special fibre; we use this to describe situations when the evaluation is constant, and situations when the evaluation is surjective. In the latter case, we describe how thi…
Following Bright and Newton, we construct an explicit K$3$ surface over the rational numbers having good reduction at 2, and for which 2 is the only prime at which weak approximation is obstructed.
Ghosh and Sarnak have studied integral points on surfaces defined by an equation x^2+y^2+z^2-xyz= m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral Hasse principle. We prove that strong approximation for integral points on any such surface, away from any finite set of places, fails, and that, for m\neq 0, 4, the Brauer group does not control strong approximation.
In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer-Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.
Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.
Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation vanish, because the abelian groups where they take values vanish. When H is connected or abelian, these algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation are the only ones, so we prove the Hasse principle and weak approximation for X under certain conditions. As an application, we obtain new suffic…
Since Poonen's construction of a variety $X$ defined over a number field $k$ for which $X(k)$ is empty and the étale Brauer--Manin set $X(\mathbf{A}_k)^\text{Br,et}$ is not, several other examples of smooth, projective varieties have been found for which the étale Brauer--Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using "Poonen's trick", i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this paper, we construct examples for which the Albane…
Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show th…
The étale Brauer\textendash Manin obstruction is equivalent to each other among Weil restrictions of an arbitrarily given quasi-projective algebraic variety defined over a number field.
In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural $2$-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi showed that over $\mathbb{C}$…
We report on our experiments and theoretical investigations concerning weak approximation and the transcendental Brauer-Manin obstruction for special Kummer surfaces.
Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a ne…
